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We present exact results for the degree distribution in a directed networkmodel that grows by node duplication (ND). Such models are useful in the studyof the structure and growth dynamics of gene regulatory networks and scientificcitation networks. Starting from an initial seed network, at each time step arandom node, a mother node, is selected for duplication. Its daughter node isadded to the network and duplicates each outgoing link of the mother node withprobability p. In addition, the daughter node forms a directed link to themother node itself. We obtain analytical results for the in-degree distribution$P_t(K_{in}=k)$, and for the out-degree distribution $P_t(K_{out}=k)$ at timet. The in-degrees follow a shifted power-law, so the network is asymptoticallyscale free. In contrast, the out-degree distribution is narrow, and convergesto a Poisson distribution in the sparse network limit and to a Gaussiandistribution in the dense network limit. Such distinction between a broadin-degree distribution and a narrow out-degree distribution is common inempirical networks such as scientific citation networks. Using this wecalculate the mean degree $\langle K_{in}\rangle_t=\langle K_{out}\rangle_t$,which converges to $1/(1-p)$ in the large network limit, for the whole range of$0<p<1$. This is in contrast to the corresponding undirected network, whichexhibits a phase transition at $p=1/2$ such that for $p>1/2$ the mean degreediverges in the large network limit. We also present analytical results for thedistribution of the number of upstream and downstream nodes from a random node.The mean values $\langle N_{up}\rangle_t=\langle N_{down}\rangle_t$ scalelogarithmically with the network size, implying that only a small fraction ofpairs of nodes are connected by directed paths, unlike the undirected ND casethat consists of a single component, hence not a small-world network.

Analytical results for the in-degree and out-degree distributions of directed random networks that grow by node duplication